Rational numbers are countable, so an order can be placed on them just like the natural numbers. Although such an ordering is not obvious (nor unique), one such ordering can be achieved by letting the x-axis represent the denominator of a rational number and the y-axis it numerator. Then starting at the origin, spiral clockwise away from it, so that each integer pair represents a rational number. Each time that the y-axis is passed, the denominator will be 0, which generates an illegal number, and all such numbers need to be skipped. Repeats, such as -1/1 and 1/-1, also occur, so only the first occurrence of a rational number should be included in the spiral. Finally, since numbers like 8/4 = 2/1 can be reduced, such repeats must also be skipped.

rational number | spiral